Laguerre's Differential Equation
I need just a little help with this question. I almost solved it except for a minor detail at the end.
Here is the question.
where m is a nonnegative interger, is called Laguerre's differential equation. Show that for each m, this equation has a polynomial solution of degree m. These polynomials are denoted by and are called Laguerre polynomials. The first few Laguerre polynomials are
Ok so I did a power series solution for the problem and arrived at
and this is where I get confused.
I see that there is a factorial pattern so I'm trying to use factorial notation to condense.
so would it be
So would the final solution be
whatever series is correct?
This solution would show depending what m is chosen that the answer would cancel out certain terms.